How does one prove the following inequality? I am currently studying Calculus and I have stumbled upon an inequality, which is crucial to one of the proofs in the “Limits of functions”, namely the following limit ($ m \in \mathbb{N} $):
$$ \lim_{x \to 0} \dfrac{\sqrt[m]{1+x} - 1}{x} = \dfrac{1}{m} $$
And the inequality I’m talking about is ($|x| > 1$):
$$ 1 - |x| < \sqrt[m]{1+x}  < 1 + |x| $$
The textbook says, that the limit immediately follows from the latter expression, but I can’t see how that is even possible, since we put $ |x| > 1 $ but in the limit we head $x$ towards 0. How should I deal with this?
 A: $$1+x=q^m$$
$$\begin{align}\lim_{x \to 0} \dfrac{\sqrt[m]{1+x} - 1}{x}
& =\lim_{q \to 1}\frac{q-1}{q^m-1}\\
&=\lim_{q \to 1}\frac{1}{1+q+q^2+\cdots+q^{m-1}}\\
&=\frac 1m.\end{align}$$
A: The inequalities cannot hold for all $|x| > 1$ over all $m \in \mathbb N_+$ because, for example, $1 + x := 1 - 2$ is not part of the natural domain of the real-valued square root function. The inequalities do indeed hold for $0 <|x| < 1$.

*

*For concreteness, the proof of the LH inequality proceeds as follows. Take the $m$-th power of both sides. Consider positive and negative values of $x$ separately.  For positive $x$, the LH inequality is obvious. For negative $x$, use $x = -|x|$ and the fact that $0<a<1$ implies $0 < a^2 < a$, and so on to $0 < a^m <a$. The proof of the RH inequality mirrors.

As for how to use the inequalities for evaluating the limit, it is for applying the squeeze theorem indirectly; the inequalities are not sharp enough for a direct attack. So probably they meant to use the inequalities to prove $\lim\limits_{x\to 0}{\sqrt[m]{1+x}} = 1$. We see $\lim\limits_{x\to 0} \operatorname{numerator}(x) = \lim\limits_{x\to 0} \operatorname{denominator}(x) = 0$, thus it might be possible to apply l'Hôpital's rule (and it turns out it is actually possible if we check).
I would prefer lone student's answer over using l'Hôpital as matter of taste. There one rationalises the numerator after a reasonable change of variables to simplify the algebra.
